Understanding Mean Relationships for Unequal Positive Numbers

The question asks about the relationship between the arithmetic mean (A) and the geometric mean (G) of two unequal positive real numbers, p and q, and a third term, which is $\frac{G^2}{A}$.

Let's first define the terms for two positive real numbers p and q:

• Arithmetic Mean (A): $A = \frac{p+q}{2}$
• Geometric Mean (G): $G = \sqrt{pq}$

For any two positive real numbers, the Arithmetic Mean is always greater than or equal to the Geometric Mean. This is known as the AM-GM inequality:

$\qquad A \ge G$

The equality ($A=G$) holds only when the numbers are equal (p = q). Since the question states that p and q are unequal positive real numbers, the strict inequality holds:

$\qquad A > G$

Now let's consider the third term given in the options: $\frac{G^2}{A}$. Let's substitute the definitions of G and A into this expression:

$\qquad \frac{G^2}{A} = \frac{(\sqrt{pq})^2}{\frac{p+q}{2}} = \frac{pq}{\frac{p+q}{2}}$

Simplifying the expression:

$\qquad \frac{pq}{\frac{p+q}{2}} = pq \times \frac{2}{p+q} = \frac{2pq}{p+q}$

The term $\frac{2pq}{p+q}$ is the definition of the Harmonic Mean (H) of two numbers p and q.

So, the third term $\frac{G^2}{A}$ is actually the Harmonic Mean (H) of p and q.

For two unequal positive real numbers, the relationship between the Arithmetic Mean (A), Geometric Mean (G), and Harmonic Mean (H) is a well-known inequality:

$\qquad A > G > H$

Substituting $\frac{G^2}{A}$ back for H, the inequality becomes:

$\qquad A > G > \frac{G^2}{A}$

This inequality tells us the correct relationship between A, G, and $\frac{G^2}{A}$ for two unequal positive real numbers.

Let's compare this result with the given options:

• Option 1: $A > G > \frac{G^2}{A}$
• Option 2: $G > \frac{G^2}{A} > A$
• Option 3: $\frac{G^2}{A} > A >G$

Our derived relationship $A > G > \frac{G^2}{A}$ matches Option 1.

Step-by-Step Analysis:

1. Identify the definitions of Arithmetic Mean (A) and Geometric Mean (G).
2. Apply the AM-GM inequality considering that the numbers are unequal ($A > G$).
3. Simplify the expression $\frac{G^2}{A}$ and recognize it as the Harmonic Mean (H).
4. Recall the relationship between AM, GM, and HM for unequal positive numbers ($A > G > H$).
5. Substitute H with $\frac{G^2}{A}$ to find the required inequality.
6. Compare the result with the given options.

The relationship $A > G > \frac{G^2}{A}$ is the correct one for two unequal positive real numbers p and q.